On the Verification of Livelock-Freedom and Self-Stabilization on Parameterized Rings

Abstract

This article investigates the verification of livelock-freedom and self-stabilization on parameterized rings consisting of symmetric, constant space, deterministic, and self-disabling processes. The results of this article have a significant impact on several fields, including scalable distributed systems, resilient and self-* systems, and verification of parameterized systems. First, we identify necessary and sufficient local conditions for the existence of global livelocks in parameterized unidirectional rings with unbounded (but finite) number of processes under the interleaving semantics. Using a reduction from the periodic domino problem, we show that, in general, verifying livelock-freedom of parameterized unidirectional rings is undecidable (specifically, Π10-complete) even for constant space, deterministic, and self-disabling processes. This result implies that verifying self-stabilization for parameterized rings of self-disabling processes is also undecidable. We also show that verifying livelock-freedom and self-stabilization remain undecidable under (1) synchronous execution semantics, (2) the FIFO consistency model, and (3) any scheduling policy. We then present a new scope-based method for detecting and constructing livelocks in parameterized rings. The proposed semi-algorithm behind our scope-based verification is based on a novel paradigm for the detection of livelocks that totally circumvents state space exploration. Our experimental results on an implementation of the proposed semi-algorithm are very promising as we have found livelocks in parameterized rings in a few microseconds on a regular laptop. The results of this article have significant implications for scalable distributed systems with cyclic topologies.